The signum function is given by $$$\mathrm{sgn}\left(\mathrm x\right)=\left\{\begin{array}{l}\frac{\mathrm x}{\left|\mathrm x\right|};\;\mathrm x\neq0\\0\;;\;\;\mathrm x=0\end{array}\right.$$$ The Fourier series expansion of sgn(cos(t)) has

The Fourier Series coefficients, of a periodic signal $$x\left( t \right),$$ expressed as $$x\left( t \right) = \sum {_{k = - \infty }^\infty {a_k}{e^{j2\pi kt/T}}} $$ are given by
$${a_{ - 2}} = 2 - j1;\,\,{a_{ - 1}} = 0.5 + j0.2;\,\,{a_0} = j2;$$
$${a_1} = 0.5 - j0.2;\,\,{a_2} = 2 + j1;\,\,$$ and
$${a_k} = 0;$$ for $$|k|\,\, > 2.$$

Which of the following is true?

Let x(t) be a periodic signal with time period T. Let y(t) = x(t - t₀) + x(t + t₀) for some t₀. The Fourier Series coefficient of y(t) are denoted by b_k. If b_k=0 for all odd k, then t₀ can be equal to

A signal $$x(t)$$ is given by
$$x\left( t \right) = \left\{ {\matrix{ {1, - {\raise0.5ex\hbox{$\scriptstyle T$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 4$}} < t \le {\raise0.5ex\hbox{$\scriptstyle {3T}$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 4$}}} \cr { - 1,{\raise0.5ex\hbox{$\scriptstyle {3T}$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 4$}} < t \le {\raise0.5ex\hbox{$\scriptstyle {7T}$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 4$}},\,\,\,} \cr { - x\left( {t + T} \right)} \cr } } \right.$$ Which among the following gives the fundamental Fourier term of $$x(t)$$?